Implicit and Iterative Methods for the Boltzmann Equation

The paper presents some approximation methods for the Boltzmann equation. In the first part fully implicit discretization techniques for the spatially homogeneous Boltzmann equation are investigated. The implicit equation is solved using an iteration process. It is shown that the iteration converges to the correct solution for the moments of the distribution function as long as the mass conservation is strictly fulfilled. For a simple model Boltzmann equation some unexpected features of the implicit scheme and the corresponding iteration process are clarified. In the second part a new iteration algorithm is proposed which should be used for the stationary Boltzmann equation. The realization of the method is very similar to the standard splitting algorithms except some new stochastic elements.